Discrete Logistic Growth

Last time, we discovered a population model of logistic growth from observing the output of an agent-based simulation. The resulting model is of the form

\[ P_{t+1} = P_{t} + rP_{t}\left(1 - \frac{P_{t}}{K}\right), \]

which deterministically determines the population size in the next generation (\( P_{t+1} \)) as a function of the current population size (\( P_{t} \)). This is known as the discrete logistic map where \( r \) is the intrinsic growth rate and \( K \) is the carrying capacity.

Question: What does it mean to have a solution to this equation, and how many solutions exist?

You might be used to solutions to algebraic equations, such as

\[ ax^2 + bx + c = 0 \]

Discuss: What does a solution to an algebraic equation look like?

Answer: In this case, the solution to the quadratic equation is zero, one or two points given by: \[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]

Discuss: What does it mean to have a solution to this equation, and how many solutions exist?

Answer: In this case, there are infinitely many solutions, sometimes referred to as a family of solutions, because they all share common traits.

You get unique solutions for this equation when you specify a starting value:

\[ P_{0} = \mathrm{a \ constant} \]

Note: There are some solutions that don't make biological sense!

A unique solution to a discrete map is an infinite sequence of points:

\[ P_{0}, P_{1}, P_{2}, P_{3}, \ldots \]

Let's simulate the equation using Desmos:

\[ P_{t} \ll K \ \ \textrm{(much smaller than)} \]

\[ \frac{P_{t}}{K} \ll 1 \]

\[ \left(1-\frac{P_{t}}{K}\right) \approx 1 \]

\[ P_{t+1} \approx (1+r)P_{t} \ \ (\textrm{Exponential growth}) \]

\[ \begin{align} P_{t+1} & = (1+r)P_{t} \\ & = (1+r)(1+r)P_{t-1}, \ \textrm{since} \ P_{t} = (1+r)P_{t-1} \\ & = (1+r)^2P_{t-1} \\ & = (1+r)^3P_{t-2}, \ \textrm{since} \ P_{t-1} = (1+r)P_{t-2} \\ & = (1+r)^?P_{0} \\ & = (1+r)^{t+1}P_{0} \end{align} \]

We have an explicit solution (exponential function)!!!

\[ P_{t} = F(t,P_{0}) = (1+r)^tP_{0} \]

What happens when \( t \rightarrow \infty \)? Depends on \( r \)!

Exponential decay when \( r < 0 \) and exponential growth when \( r > 0 \). What about \( r = 0 \)?

\[ P_{t} \approx \frac{K}{2} \Longrightarrow \left(1-\frac{P_{t}}{K}\right) \approx \frac{1}{2} \]

\[ P_{t+1} \approx P_{t} + \frac{r}{2}P_{t} = \left(1+\frac{r}{2}\right)P_{t} \]

\[ P_{t} \approx K \Longrightarrow \left(1-\frac{P_{t}}{K}\right) \approx 0 \]

\[ P_{t+1} \approx P_{t} + rP_{t}\cdot 0 = P_{t} = K \]

\( K \) is called a fixed point.

There are a few ways to study a discrete map like this.

• Simulate in time using a computer
  • Pro: Easy to do; Con: Not rigorous
• Explicitly solve the equation, i.e. convert to a form \( P_{t} = F(t, P_{0}) \)
  • Pro: If exists, know everything; Con: Very few models can be explicitly solved.
• Study the behavior of solutions (without solving them!)

For the discrete logistic model, even though it is relatively simple, there does not exist an explicit solution!

We've simulated using a computer - what about exploring behavior of solutions?? Dynamical Systems Theory!!

In dynamical systems theory, one of the most important questions is:

Question: What happens to solutions as \( t \rightarrow \infty \)?

We saw this in the simulations - there is a transient period (\( t < \infty \)) where the solution seems to converge to an equilibrium (\( t \rightarrow \infty \)).

The simplest example of equilibrium solutions are known as fixed points. Let's explore this idea through the simulations.

Definition: Fixed points of a discrete map are denoted by \( P_{\infty} \) and satisfy the relation

\[ P_{t} = P_{\infty}, \ \textrm{for all $t$} \]

This also means \( P_{t+1} = P_{t} = P_{\infty} \). So how do we find fixed points for a discrete map?

If the discrete map is given by

\[ P_{t+1} = F(P_{t}) \]

then fixed points satisfy the equation

\[ P_{t+1} = {\bf F(P_{t}) = P_{t}} \Rightarrow F(P_{t})-P_{t} = 0 \]

In other words, fixed points simultaneously solve (algebraically) the two equations:

\[ \begin{align} P_{t+1} & = F(P_{t}) \\ P_{t+1} & = P_{t} \end{align} \]

We can visual the dynamics AND these two equations in a cobweb plot.

To do this, we will (graphically) assign \( y = P_{t+1} \) and \( x = P_{t} \) to give

\[ \begin{align} y & = F(x) \\ y & = x \end{align} \]

Let's plot these two equations.

\[ \begin{align} P_{t+1} & = P_{t} + rP_{t}\left(1 - \frac{P_{t}}{K}\right) \\ P_{t+1} & = P_{t} \end{align} \]

\( \Longrightarrow \)

\[ \begin{align} P_{t} & = P_{t} + rP_{t}\left(1 - \frac{P_{t}}{K}\right) \\ 0 & = rP_{t}\left(1 - \frac{P_{t}}{K}\right) \\ \end{align} \]

\( \Longrightarrow \ P_{t} = 0 \ \) (or) \( \ 1 - \frac{P_{t}}{K} = 0 \ \Longrightarrow \ P_{t} = 0 \ \) (or) \( \ P_{t} = K \).

Definition: A fixed point is called stable (asymptotically stable) if all small deviations from the fixed point converge/limit back to the fixed point as \( t\rightarrow\infty \).

A fixed point is called unstable if all small deviations from the fixed point DO NOT converge back to the fixed point as \( t\rightarrow\infty \).

Question: How do we determine the stability of fixed points?

Definition: A fixed point \( P_{\infty} \) is stable when

\[ \left|F^{\prime}(P_{\infty})\right| < 1 \]
A fixed point \( P_{\infty} \) is unstable when

\[ \left|F^{\prime}(P_{\infty})\right| > 1 \]

Given that \[ F(P_{\infty}) = P_{\infty} + rP_{\infty} - \frac{r}{K}P_{\infty}^2 \] we have \[ \begin{align} F^{\prime}(P_{\infty}) & = 1 + r - \frac{2r}{K}P_{\infty} \end{align} \] Thus, we have a stable fixed point when

\[ \begin{align} & \left|F^{\prime}(P_{\infty})\right| < 1 \\ \Longrightarrow & \left|1+r-\frac{2r}{K}P_{\infty}\right| < 1 \end{align} \]

Thus, we have a stable fixed point when

\[ \begin{align} & \ \left|F^{\prime}(P_{\infty})\right| < 1 \\ \Longrightarrow & \ \left|1+r-\frac{2r}{K}P_{\infty}\right| < 1 \end{align} \]

\( P_{\infty} = 0 \) is stable when

\[ \begin{align} & \ \left|1+r\right| < 1 \\ \Longrightarrow & \ -1 < 1+r < 1 \\ \Longrightarrow & \ -2 < r < 0 \end{align} \]

Thus, we have a stable fixed point when

\[ \begin{align} & \ \left|F^{\prime}(P_{\infty})\right| < 1 \\ \Longrightarrow & \ \left|1+r-\frac{2r}{K}P_{\infty}\right| < 1 \end{align} \]

\( P_{\infty} = K \) is stable when

\[ \begin{align} & \ \left|1+r-2r\right| < 1 \\ \Longrightarrow & \ \left|1-r\right| < 1 \\ \Longrightarrow & \ \left|r-1\right| < 1 \\ \Longrightarrow & \ -1 < r-1 < 1 \\ \Longrightarrow & \ 0 < r < 2 \\ \end{align} \]

“In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable values with a solid line and unstable values with a dotted line, although often the unstable points are omitted. Bifurcation diagrams enable the visualization of bifurcation theory.”
- Wikipedia

Note: \( r \) values off by 1!!

Note: \( r \) values off by 1!!